A double Atwood's machine consists of two massless pulleys, each of radius \(R\), some massless string, and three weights, with masses \(m_{1}, m_{2}\), and \(m_{3}\). The axis of pulley 1 is supported by a strut from the ceiling. A piece is string of length \(\ell_{1}\) is slung over the pulley, and one end of the string is attached to weight \(m_{1}\) while the other end is attached to the axis of pulley 2. A second string of length \(\ell_{2}\) is slung over pulley 2 ; one end is attached to \(m_{2}\) and the other to \(m_{3}\). The strings are inextendible, but otherwise the weights and pulley 2 are free to move vertically. Let \(x\) be the distance of \(m_{1}\) below the axis of pulley 1, and \(y\) be the distance of \(m_{2}\) below the axis of pulley 2.

(a) Find the Lagrangian \(L(x, y, \dot{x}, \dot{y})\).

(b) Find the canonical momenta \(p_{x}\) and \(p_{y}\), in terms of \(\dot{x}\) and \(\dot{y}\).

(c) Find the Hamiltonian of the system in terms of \(x, y, \dot{x}\), and \(\dot{y}\). (Note that to go on and find the motion of the system using Hamilton's equations, one must first write \(H\left(x, y, p_{x}, p_{y}\right)\), without \(\dot{x}\) and \(\dot{y}\). This step, and the next step of solving the equations, involves a lot of algebra. This illustrates the fact that in somewhat complicated problems one could long since have written out Lagrange's equations and solved them, by the time one has even written out the Hamiltonian in canonical form.)